Position-based Matching with Multi-Modal Preferences

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**Read paper on the following link:** https://ifaamas.org/Proceedings/aamas2022/pdfs/p1373.pdf **Abstract:** Many models have been proposed for computing a one-to-one matching between two equal-sized sets/sides of agents, each assigned with one preference list of the agents in the opposite side. The most prominent one might be the Stable Matching model. Recently, the Stable Matching model has been extended to the multi-modal setting \cite{UncertainLinearPfAziz2020,ChenEC2018Multi,Miyazaki19Jointly}, where each agent has more than one preference lists, each represents a criteria based on which the agents of the opposite side are evaluated. We use a layer to denote the set of preference lists of agents, which are based on the same criteria. Thus, the single modal matching problem has only one layer. This setting finds applications in many real-world scenarios. However, it turns out that computing a stable matching with multi-modal preferences is NP-hard and W-hard with respect to several natural parameters. Here, we introduce three position-based matching models, which minimize the ``dissatisfaction score''. We define four dissatisfaction scores, namely, the one of a single agent (the position of an agent's matching partner in his/her preference), the one of a matching pair (the sum of the dissatisfaction scores of two agents forming a matching pair), the dissatisfaction score of one side (the sum of the dissatisfaction scores of all agents in one side), and the total dissatisfaction score (the sum of all dissatisfaction scores of all agents). The first model minimizes the total respective dissatisfaction score over all layers, while the second minimizes the maximum of the respective score over all layers. The third model seeks for a matching $M$ which is \PO, meaning that there does not exist a matching $M^\prime$, which is at least as good as $M$ with respect to the respective dissatisfaction score in all layers, but is strictly better in at least one layer. We present diverse complexity results for these three models, among others, polynomial-time solvability for the first model. We also investigate the generalization which given an upper bound on the dissatisfaction score, computes a matching involving subsets of agents and a subset of layers. Hereby, we mainly focus on the parameterized complexity with respect to parameters such as the size of agent subsets, or the size of the layer subset and achieve fixed-parameter tractability as well as intractability results.